Statistics Papers

Document Type

Journal Article

Date of this Version

1986

Publication Source

The Annals of Probability

Volume

14

Issue

3

Start Page

877

Last Page

890

DOI

10.1214/aop/1176992444

Abstract

Let F and G be two continuous distribution functions that cross at a finite number of points − ∞ ≤ t1 < ⋯ < tk ≤ ∞. We study the limiting behavior of the number of times the empirical distribution function Gn crosses F and the number of times Gn crosses Fn. It is shown that these variables can be represented, as n → ∞, as the sum of k independent geometric random variables whose distributions depend on F and G only through F′(ti)/G′(ti), i = 1, …, k. The technique involves approximating Fn(t) and Gn(t) locally by Poisson processes and using renewal-theoretic arguments. The implication of the results to an algorithm for determining stochastic dominance in finance is discussed.

Keywords

asymptotic distribution, boundary crossing probability, geometric distribution, Poisson process, renewal theory, stochastic dominance algorithm, Weiner-Hopf technique

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Date Posted: 27 November 2017

This document has been peer reviewed.